Cohen-Macaulay seminormalizations of unions of linear subspaces

Abstract

This paper is concerned with the unions of subspaces of an afIine space regarded as affine varieties. Let R denote the co-ordinate ring of such a union. Then according to a theorem of Orecchia’s [S], the seminormaliza- tion S of R can be represented as the section ring (or pull-back) of the sheaf d of the co-ordinate rings on the intersection poset X of the subspaces. In this paper, we give a homological condition on X and d equivalent to S being Cohen-Macaulay (Theorem 4.3). Two particular cases of this result have been known before. The Reisner Theorem [6] covered the case of co-ordinate subspaces (when R = S is the face ring of a complex) and Geramita and Weibel [2] soIved the problem for planes. The main idea of the proof is to construct a resolution C of S consisting of S-modules of a known depth and then estimate the depth of S by the length of C. This idea was applied in [7] to the section rings of flasque sheaves. Since in this paper the sheaves are not necessarily flasque, the con- struction of C is a bit more complicated and involves the local Tech cohomology of X with coefficients in &. A drawback of the method is that it gives very little information about R itself when it is not seminormal. This paper is organized as follows. In Section 1, the terminology and background of sheaves on posets is given, In Section 2, we study the (Tech) cohomology of a covering of a poset with coefficients in a sheaf. The next section is technical. In it we compare the depth of the rings of sections on certain closed subsets with their depth as modules over the ring of global sections. In Section 4, we prove the main result of the paper. Finally, Section 5 contains examples and some unsolved problems.